A student conducted an experiment and found that an ideal gas obeys the condition $\large\frac{p^2}{\rho} = $ constant. Initially the gas is at a temperature $T$, pressure $p$ and density $\rho$. The gas expands such that the density changes to $\rho/2$, which of the following is true?

$(A): $ the pressure of the gas changes to $\sqrt (2p)$
$(B): $ the pressure of the gas is unchanged
$(C): $ the graph of the above process on the P-T graph is a parabola
$(D): $ the graph of the above process on the P-T graph is a hyperbola